Two-boundary problems for a Poisson process with exponentially distributed component

For a Poisson process with exponentially distributed negative component, we obtain integral transforms of the joint distribution of the time of the first exit from an interval and the value of the jump over the boundary at exit time and the joint distribution of the time of the first hit of the interval and the value of the process at this time. On the exponentially distributed time interval, we obtain distributions of the total sojourn time of the process in the interval, the joint distribution of the supremum, infimum, and value of the process, the joint distribution of the number of upward and downward crossings of the interval, and generators of the joint distribution of the number of hits of the interval and the number of jumps over the interval. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 7, pp. 922–953, July, 2006.     

Optimization of axisymmetric membrane shells

Problems of the joint optimization of the shape and distribution along the meridian of the thickness of membrane shells of revolution under the action of axisymmetric loads are considered, taking account of the constraints concerning the strength of the shell and the volume of its cavity. General formulations of problems of the optimal design of shells of revolution are given and the optimal shape of a shell and the corresponding thickness distribution are investigated. Results of the exact solution of problems of the optimal design of closed shells of revolution when there is an internal pressure are presented. The simultaneous introduction of two control functions, describing the shape of the shell and the distribution of its thickness, not only ensures a substantial reduction in the mass of a shell but also leads to significant mathematical simplifications, which enable the solution of the optimization problem being considered to be obtained in an analytical form.     

基于加总和乘积的方差分解法及其应用

研究了加总式和乘积式的方差分解问题,证明了在因变量等于各自变量之和的条件下,因变量方差等于各自变量与因变量的协方差之和;在因变量等于各自变量之乘积的条件下,因变量对数值的方差等于各自变量对数值与因变量对数值的协方差之和．以中国31个省份2005－2012年的居民人均收入及其影响因素的统计数据资料为例,说明了加总式和乘积式的方差分解法的具体应用．     

Geometry of a five link mechanism with two degrees of freedom

The paper is devoted to the solution of straight and inverse geometrical tasks of five link mechanism with two degrees of freedom. The solution of the mentioned problem is very important in order to determine kinematic parameters of actuators. The problem can be divided into two parts. The first part is considered when we are given the coordinates of the output link of the mechanism and the necessity arises of determining the angles of rotation of actuators. On the other hand, it is very important to determine the position of the output link when the angles of rotation of the actuators are known. Here we consider that the mechanism is composed only of five classes of rotating kinematic pairs and the actuators are situated at the junctions of frames and links of the examined mechanism. The solution of the said problem is based on utilization of homogenous coordinates. On the basis of the obtained equations of motion, one can calculate the trajectories of motion of the output link as well angles of rotation of the actuators by taking into consideration preliminary given kinematic parameters of the mechanism. Here we also obtain equations for calculating values of speed and acceleration of the links of the mechanism. The calculations differ from known methods in simplicity and high performance, which would be useful for programming actuators mounted in the joints of the linkage.     

区间数的绝对值与区间值函数的极限

研究了区间数的绝对值和区间值函数的极限问题.首先,讨论了区间数的H-差的性质,得到了H-差的两个运算法则;然后,给出了区间数的绝对值概念,并讨论了区间数绝对值的性质;最后,借助区间数的H-差和绝对值的概念,建立了区间值函数极限概念的一种新的表达方式,给出了极限存在的充分必要条件,证明了极限值的唯一性及对加法运算和数乘运算的封闭性.     

Algorithm for computing the mechanical impedances of reinforcing ribs of various shapes

The problem is considered of finding the mechanical impedances of reinforcing ribs of various shapes. The rib is modeled by a narrow plate. The thickness of the rib (the thickness of the plate from which the rib is made) is thus considered small as compared with the height of the rib (the width of the plate). The height of the rib h is assumed small as compared with the lengths of waves propagating in the material of the plate. One of the edges of the rib is assumed to be free of stresses. The matrix giving the relation between the stresses and the velocities of displacements on the second (loaded) edge is called the impedance matrix. An algorithm is proposed for computing the coefficients of the expansion of the impedance matrix in a Taylor series in powers of h. The method is illustrated by the computation of the mechanical impedances of rectilinear and annular reinforcing ribs.     

Complexity Estimation for an Algorithm of Searching for Zero of a Piecewise Linear Convex Function

It is known that the problem of the orthogonal projection of a point to the standard simplex can be reduced to solution of a scalar equation. In this article, the complexity is analyzed of an algorithm of searching for zero of a piecewise linear convex function which is proposed in [30]. The analysis is carried out of the best and worst cases of the input data for the algorithm. To this end, the largest and smallest numbers of iterations of the algorithm are studied as functions of the size of the input data. It is shown that, in the case of equality of elements of the input set, the algorithm performs the smallest number of iterations. In the case of different elements of the input set, the number of iterations is maximal and depends rather weakly on the particular values of the elements of the set. The results of numerical experiments with random input data of large dimension are presented.     

On subcritical development of a shear crack in a composite with viscoelastic components

We present results of an investigation of the development of a transverse shear crack in a composite material with linearly viscoelastic components under external shear load. The solution is divided into the following two main stages: determination of the time dependence of the crack tip opening displacement and determination of the crack-growth kinetics as a result of the solution of integral equations. In the first stage, we use the solution of the corresponding elastic problem of determination of the crack opening displacement and the problem of determination of the effective moduli of the composite reinforced with unidirectional discrete fibers. Using the theoretically proved principle of elasto-viscoelastic analogy and the method of Laplace inverse transformation, we obtain a solution in a time domain. In the second stage, using the criterion of critical crack opening displacement for a transverse shear crack and an equation for the viscoelastic crack opening displacement of this crack, we construct an equation of crack growth. We present results of the numerical solution, which illustrate the influence of relations between the relaxation parameters of the materials of the components on the durability of the body with a crack.     

Asymptotics of a Solution of a Three-Dimensional Nonlinear Wave Equation near a Butterfly Catastrophe Point

We consider an optimal distributed control problem in a planar convex domain with smooth boundary and a small parameter at the highest derivatives of an elliptic operator. The zero Dirichlet condition is given on the boundary of the domain, and the control is included additively in the inhomogeneity. The set of admissible controls is the unit ball in the corresponding space of square integrable functions. Solutions of the obtained boundary value problems are considered in the generalized sense as elements of a Hilbert space. The optimality criterion is the sum of the squared norm of the deviation of the state from a given state and the squared norm of the control with a coefficient. This structure of the optimality criterion makes it possible to strengthen, if necessary, the role of either the first or the second term of the criterion. In the first case, it is more important to achieve the desired state, while, in the second case, it is preferable to minimize the resource consumption. We study in detail the asymptotics of the problem generated by the sum of the Laplace operator with a small coefficient and a first-order differential operator. A feature of the problem is the presence of the characteristics of the limit operator which touch the boundary of the domain. We obtain a complete asymptotic expansion of the solution of the problem in powers of the small parameter in the case where the optimal control is an interior point of the set of admissible controls.     

A Note on the Radical of a Module Category

We characterize the finiteness of the representation type of an artin algebra in terms of the behavior of the projective covers and the injective envelopes of the simple modules with respect to the infinite radical of the module category. In case the algebra is representation-finite, we show that the nilpotency of the radical of the module category is the maximal depth of the composites of these maps, which is independent from the maximal length of the indecomposable modules.     

Asymptotic solutions of coupled dynamic problems of thermoelasticity for isotropic plates

The asymptotic method of solving boundary-value problems of the theory of elasticity for anisotropic strips and plates is used to solve coupled dynamic problems of thermoelasticity for plates, on the faces of which the values of the temperature function and the values of the components of the displacement vector or the conditions of the mixed problem of the theory of elasticity are specified. Recurrence formulae are derived for determining the components of the displacement vector, the stress tensor and for the temperature field variation function of the plate.     

Analytical determination of time-dependent functions of drifts of a triaxial gyrostabilizer

Equations which describe the time-dependence of the vector of the instantaneous velocity of rotation of a triaxial gyrostabilizer (TGS) platform located on Earth, with respect to the gyroscopical system of coordinates, are obtained within the framework of the precession theory at an arbitrary polynomial dependence of the rate of the drift on overloads. Based on the properties of the TGS and the details of its use we discuss the approximation of these equations from which the analytical time-dependent function of the drift rate of a gyrostabilized platform is found. The results obtained significantly simplify the procedure of solving the problems of identification of the TGS's parameters and estimating its current orientation relative to the specified system of coordinates in inertial navigation systems.     

The error and guaranteed accuracy of cubature formulas in multidimensional periodic Sobolev spaces

We give an upper bound for the deviation of the norm of a perturbed error from the norm of the original error of a cubature formula in a multidimensional bounded domain. The deviation arises as a result of the joint influence on the computations of small variations of the weights of a cubature formula and rounding in the subsequent calculations of the cubature sum in the given standards (formats) of approximation to real numbers. We estimate the practical error of a cubature formula acting on an arbitrary function from the unit ball of a normed space of integrands. The resulting estimates are applied to studying the practical error of cubature formulas in the case of integrands in Sobolev spaces on a multidimensional cube. The norm of the error in the dual space of the Sobolev class is represented as a positive definite quadratic form in the weights of the cubature formula. We estimate the practical error for cubature formulas constructed as the direct product of quadrature formulas of rectangles along the edges of the unit cube. The weights of this direct product are positive.     

Estimate of the Order of Growth of a Class of Entire Functions on the Real Axis

For a class of entire functions we study the problem of estimation of the order of growth of functions on the real axis. This problem is important for the justification of the integral representation of bounded solutions to certain partial differential equations considered in other papers of the authors. In order to obtain an estimate of the order of growth of a function on the real axis, we use the method of differential equations. The method is based, on one hand, on the construction of a system of first-order ordinary differential equations whose solution is a vector function of traces of function and its derivatives on the real axis. On the other hand, under the respective change of variables in the system of equations, we obtain an estimate of the solution to the system of equations for a large positive values of the argument. The obtained estimate is non-trivial and shows the way a complex parameter of a power series affects the order of growth of a function.     

The dynamics of pyramidal bodies within the framework of the local interaction model

Two-dimensional inertial motion of pyramidal bodies in a medium is investigated, on the assumption that the force exerted by the medium on their surface is described by the local interaction model. Assuming unseparated flow around the bodies and small perturbations applied at the initial time to the parameters of rectilinear motion, an analytical solution is constructed of the problem of the two-dimensional motion of slender bodies with bases whose contour is a rhombus or a star consisting of four symmetrical cycles. It is shown that the solution provides the basis for a complete parameterc analysis of the dynamics of the body and for evaluating the forces and torques experienced by the body along its trajectory. A criterion for the stability of the body is found, using which, knowing the velocity, mass and position of the body's centre of gravity, one can determine the form of the perturbed motion of the pyramidal body. It is shown that the body shape is one of the most important factors affecting the stability of motion, and that, of all bodies with the same shape and position of the centre of mass, those with the least mass have the largest reserve of stability. The analytical results are confirmed by numerical solution of the Cauchy problem for the system of equations of motion obtained without the simplifying assumptions.     

The problem of the equilibrium of a helical spring in the non-linear three-dimensional theory of elasticity

The problem of the loading of a helical spring by an axial force and a torque is considered using the three-dimensional equations of the non-linear theory of elasticity. The problem is reduced to a two-dimensional boundary-value problem for a plane region in the form of the transverse cross section of the coil of the spring. The solution of the two-dimensional problem obtained enables the equations of equilibrium in the volume of the body and the boundary conditions on the side surface to be satisfied exactly. The boundary conditions at the ends of the spring are satisfied in the integral Saint-Venant sense. The problem of the equivalent prismatic beam in the theory of springs is discussed from the position of the solution of the non-linear Saint-Venant problem obtained. The results can be used for accurate calculations of springs in the non-linear strain region, and also when developing applied non-linear theories of elastic rods with curvature and twisting.     

The calculation of the energy losses in a sliding elastic contact with friction and wear (the plane problem)

The plane problem of the sliding contact of a punch with an elastic foundation when there is friction and wear is considered. Assuming the existence of a steady solution in a moving system of coordinates, relations are derived between the sliding velocity, the wear, the contact stresses and the displacements for an arbitrary dependence of the wear rate on the contact pressure. Taking into account the presence of a deformation component of the friction force, an equation is written for the balance of the mechanical energy for the punch - elastic base system considered. It is shown that the equality of the work of the external force in displacing the punch to the losses due to friction and the change in the shape of the foundation due to wear is satisfied when the work done by the contact stresses on the increments of the boundary displacements is equal to zero, and the frictional losses must be determined taking into account the non-uniformity of the distributions of the shear contact stresses and the sliding velocity in the contact area. Two special cases of the foundation in the form of a wide and narrow strip are considered, for which the total coefficient of friction is calculated, taking into account the deformation component of the friction force.     

Optimal control of the rotation of a system of two bodies connected by an elastic rod

Optimal controls of the rotation of a mechanical system consisting of two rigid bodies, joined by an elastic rod, through a specified angle about an axis passing through the centre of mass of one of the bodies are constructed. The problem of the optimal control of the rotation of the system through a given angle with complete suppression of the oscillations of the elastic rod at the minimum of the energy functional of the control moment and the problem of time-optimality for a specified constraint on the energy functional of the control moment are solved.     

The stability of a ring under the action of a linear torque,constant along the perimeter

Exact analytical solutions of problems on the static and dynamic forms of the loss of stability of a ring, under the action of a linear torque constant along the perimeter, are found using the consistent equations of the theory of plane curvilinear rods constructed earlier taking account of transverse shears. Two forms of torsion of the ring are examined: the external forces creating a torque remain in the plane of a cross-section of the ring in its initial undeformed state (“dead” forces, case 1) or in its deformed state (“follower” forces, case 2). It is shown that, in the second case, the solution of the static instability problem found is practically identical to the solution of the problem corresponding to the dynamic formulation and is reduced to an examination of the oscillations about the static equilibrium position. In the case of both forms of loading, loss of stability of the ring occurs without deformation of its axial line, with it bending predominantly in the plane of the ring accompanied by a slight distortion. It is established that a study of the forms of loss of stability of the ring for the type of loading considered is only possible using the equations constructed, taking account of transverse shear.     

A criterion for the growth of cracks with bonds in the end zone

A fracture criterion which takes account of the work done in the deformation of bonds in the end zone of a crack is proposed for analysing the quasistatic growth of a crack with bonds in the end zone. The energy condition that the deformation energy release rate at the crack tip is equal to the rate of deformation energy consumption by the bonds in the end zone of the crack (the first fracture condition) corresponds to the state of limit equilibrium of the crack tip. The rupture of bonds at the trailing edge of the end zone is determined by the condition for their limiting traction (the second fracture condition). Starting from these two conditions, the processes of subcritical and quasistatic crack growth are considered for the case of a rectilinear crack at interface of materials and the two basic fracture parameters, the critical external load and the size of the end zone of the crack in the state of limit equilibrium, are determined. Analytical expressions are obtained for the deformation energy release rate at the crack tip and the rate of deformation energy consumption by the bonds and, also, the dependences of the critical external load and size of the end zone of the crack on the crack length in the case of a rectilinear crack in a homogeneous body with bond tractions which are constant and independent of the external load. The limit cases of a crack which is filled with bonds and a crack with a short end zone are considered.